Buckling and post-buckling of extensible rods revisited: A multiple-scale solution

The conditions for an overhanging flexible strip to slide off a flat surface are investigated. This problem rrmy be applicable to pieces of paper, fabric, leather, and other flexible materials, including plastic and metallic strips used herein for experimental comparisons. The critical overhang length depends on (a) the length, weight per unit length, and bending stiffness of the strip, (b) the coefficients of friction {CoFs) between the strip and both the surface and its edge, and (c) the inclination of the surface. The strip is modeled as an inextensible elástica. A shooting method is applied to solve the nonlinear equations that are based on equilibrium, geometry, and Coulomb friction. Three types of equilibrium shape are obtained. In the most common type, one end of the strip overhangs the edge and the other end contains a segment that is in contact with the surface. In another type, contact only occurs at the nonoverhanging end and at the edge. The third type involves the strip balancing on the edge of the surface. The ratio of the critical overhang length to the total strip length is plotted as a function of the surface CoF edge CoF, and weight parameter for a horizontal surface, ln most cases, this ratio increases as the CoFs and the strip's bending stiffness increase, and decreases as the strip's weight per unit length increases. The rotation of the strip at the edge tends to increase as the strip's weight per unit length, the strip's length, and the surface CoF increase, and to decrease as the strip's bending stiffness increases. Inclined surfaces are alio considered, and the critical overhang length decreases as the surface slopes more downward toward the edge. The theoretical results are compared with e.xperimental data, and the agreement is good.

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“…The weight parameter w defined in Eq. (8), along with the coefficients of friction ß^. and ß^, determines when such failure occurs.…”

Section: Discussionmentioning

confidence: 99%

“…WÛ El ' (8) 'J=h2) Numerical solutions are obtained by a shooting method using the subroutines NDSolve and FindRoot in MATHEMATICA [5]. For shape I, the conditions at D, where S[=S2=0, are x¡=y,=X2=y2 =0, 6^ = 62, and m|=-m2.…”

Section: Formulationmentioning

confidence: 99%

Instability of Flexible Strip Hanging Over Edge of Flat Frictional Surface

Plaut

Dillard

2011

Journal of Applied Mechanics

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“…Magnusson et al [8] abandoned the commonly adopted assumption in the investigation of the post-buckling behaviour of compressed columns, of constant loading force in post-buckling equilibrium states, and investigated instead the behaviour of "extensible" elastica. Their study was further enhanced by the work of Mazzilli [9] who analytically approached the problem of extensible elastica by loading not only axial forces but also transverse forces and bending moments at the column ends, and found post-buckled configurations in a number of cases.…”

Section: Introductionmentioning

confidence: 99%

Buckling and Post-buckling Behaviour of Prismatic Aluminium Columns Submitted to a Series of Compressive Loads

Ziółkowski

Imiełowski

2010

Exp Mech

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The buckling and post-buckling behaviour of prismatic aluminium columns from stocky to very slender shapes is investigated. The unconventional, in terms of buckling tests, displacement control of compressive load and a series of loadings provided an enhanced insight into the buckling process. A phenomenon of buckling load drop has been detected in columns of intermediate slenderness, reaching over 20% of the load early critical value. This newly observed occurrence resembles finite disturbance instability, which until recently was commonly believed to only appear in cases of thin walled cylindrical shells, but not columns. The observation is in contradiction to predicted results from the elasto-plastic buckling models of Engesser or Shanley, with constant or growing values of load during the post-buckling process. Further tests on columns of intermediate slenderness, with strain gauges glued at node and anti-node locations of the buckled profiles, revealed that even minute buckling results in fields of highly non-symmetric residual microplastic strain. The results of the present study indicate that running column buckling tests under displacement control is worthy of being adopted as common practice. The envelope of column post-buckling states can be conveniently determined. This information will in turn allow for the quick and reliable estimation of the safety of a column, which has undergone accidental or deliberate damage in the form of limited buckling when under operational load.

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“…Lacarbonara (2008) constructed the post-buckling solutions of non-uniform linearly and nonlinearly elastic rods via a higher-order perturbation approach. Mazzilli (2009) dealt with post-buckling of all five classical kind of Euler buckling with multiple scale solution and pursued the critical load and post-buckling configuration. Shavartsman (2009) presented a direct method for analysis of flexible cantilever beam subjected to two follower forces.…”

Section: Introductionmentioning

confidence: 99%

Post-buckling analysis of variable cross-section cantilever beams under combined load via differential quadrature method

2010

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Based on the geometrically nonlinear theory of extensible elastic rods, the governing equations for post-buckling of variable crosssection cantilever beams subjected to a concentrated axial load at its free end and a non-uniformly distributed axial load are formulated. The Differential Quadrature Method (DQM) as a simple and computationally efficient numerical method is used to obtain the critical buckling load. The accuracy of the method is verified by comparing the results with those of the analytical result from the elliptical integral, and other methods such as shooting method or multiple scale method. The post buckling configuration of the beam is obtained for different loading conditions.

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Buckling and post-buckling of extensible rods revisited: A multiple-scale solution

Cited by 32 publications

References 8 publications

Instability of Flexible Strip Hanging Over Edge of Flat Frictional Surface

Instability of Flexible Strip Hanging Over Edge of Flat Frictional Surface

Buckling and Post-buckling Behaviour of Prismatic Aluminium Columns Submitted to a Series of Compressive Loads

Post-buckling analysis of variable cross-section cantilever beams under combined load via differential quadrature method

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